Converses to generalized Conway--Gordon type congruences

TL;DR

This paper extends Conway-Gordon type congruences for spatial complete graphs, showing that all integers of a certain form can be realized as sums of Conway polynomial coefficients over Hamiltonian knots.

Contribution

It proves that every integer of the form (n-5)! q + r_n can be realized as a sum of Conway polynomial coefficients in some spatial complete graph on n vertices.

Findings

All integers of the form (n-5)! q + r_n are realizable.

Generalization of Conway-Gordon congruences to a converse statement.

Applicable to spatial complete graphs with n ≥ 7 vertices.

Abstract

It is known that for every spatial complete graph on $n\ge 7$ vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to $r_{n}$ modulo $(n-5)!$, where $r_{n} = (n-5)!/2$ if $n=8k,8k+7$, and $0$ if $n\neq 8k,8k+7$. In particular the case of $n=7$ is famous as the Conway--Gordon $K_{7}$ theorem. In this paper, conversely, we show that every integer $(n-5)! q + r_{n}$ is realized as the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots in some spatial complete graph on $n$ vertices.